A Dynamical Uncertainty Principle in von Neumann Algebras by Operator Monotone Functions

Abstract

Suppose that A 1 , . . . , A N are observables (selfadjoint matrices) and ρ is a state (density matrix). In this case the standard uncertainty principle, proved by Robertson, gives a bound for the quantum generalized variance, namely for det{Cov ρ (A j , A k )}, using the commutators [A j , A k ]; this bound is trivial when N is odd. Recently a different inequality of Robertson-type has been proved by the authors with the help of the theory of operator monotone functions. In this case the bound makes use of the commutators [ρ, A j ] and is non-trivial for any N . In the present paper we generalize this new result to the von Neumann algebra case. Nevertheless the proof appears to simplify all the existing ones.

On a correspondence between regular and non-regular operator monotone functions

“…The Wigner-Yanase-Dyson (WYD) measures of information are not only used in quantum information theory. A list of applications in other fields includes: i) strong subadditivity of entropy [18,17]; ii) homogeneity of the state space of factors of type III 1 [6]; iii) measures for quantum entanglement [4,15]; iv) uncertainty relations (see [1,9] and references therein); v) hypothesis testing [2]. This is, in a certain sense, not surprising since the WYD-information is connected to special choices of quantum Fisher information (see [13,10]).…”